In this paper, we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to non-linear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold. This allows one to define, in addition to the two geometric structures, a Weitzenböck connection on the material manifold. We use this connection to express natural uniformity in a geometrically meaningful way. The concept of uniformity is then extended to the Riemannian and Euclidean structures. Finally, we discuss the role of non-uniformity in the form of material forces that appear in the configurational form of the balance of linear momentum with respect to the two structures.

中文翻译：

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非弹性中的黎曼和欧几里德材料结构

在本文中，我们讨论了关于材料流形上的黎曼和欧几里得几何结构的非弹性体力学。这两种结构提供了两组等效的控制方程，它们对应于非线性非弹性的几何方法和经典方法。本文提供了两种方法之间的并行性，并解释了如何从一种方法转到另一种方法。我们将变形梯度的乘法分解设置为材料流形中框架的非完整变化。除了两个几何结构之外，这还允许定义材料歧管上的 Weitzenböck 连接。我们使用这种连接以几何意义的方式表达自然均匀性。然后将均匀性的概念扩展到黎曼和欧几里德结构。最后，我们讨论了非均匀性在物质力的形式中的作用，这些力以两种结构的线性动量平衡的配置形式出现。